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Computer Methods in Applied Mechanics and Engineering

Padded continued fraction absorbing boundary conditions for dispersive waves

Abstract

Continued fraction absorbing boundary conditions (CFABCs) are new arbitrarily high-order local absorbing boundary conditions that are highly effective in simulating wave propagation in unbounded domains . The current versions of CFABCs are developed for the non-dispersive acoustic wave equation with convex polygonal computational domains. In this paper, the CFABCs are modified through augmentation of special padding elements, and are effective for absorbing evanescent waves occurring in dispersive wave problems while retaining their absorption properties for propagating waves. The padded CFABCs for dispersive wave equations result in fourth order evolution equations, which are solved using an efficient combination of Crank Nicholson method, Newmark time-stepping scheme, and operator splitting ideas. The effectiveness of the padded CFABCs and their implementation is illustrated through numerical examples with varying levels of dispersion.

Introduction

Computational modeling of wave propagation in unbounded domains often involves truncating the domain around the region of interest (referred to as the interior), and applying absorbing boundary conditions (ABCs) on the truncation boundary. An effective absorbing boundary condition should accurately simulate the wave radiation into the exterior, while keeping the computational cost low. Many researchers have proposed several ABCs since the 1970s, which can be classified into two broad classes: differential-equation-based and material-based [1]. Differential-equation-based ABCs are obtained by factoring the wave equation into outward and inward propagating operators and permitting only outgoing waves by eliminating the inward propagation operator. Material-based ABCs, on the other hand, are realized by surrounding the computational domain with a lossy material that dampens the outgoing waves. Differential-equation-based ABCs can be further classified into two subcategories: exact (global) ABCs and approximate (local) ABCs. Computation with global absorbing boundaries involves obtaining the exterior Green's function and coupling it with the interior, which involves expensive convolution operations [2]. Global ABCs are useful for small-scale wave propagation problems and lead to highly accurate results. In spite of many innovations, they still tend to be computationally expensive for large-scale problems. With large-scale problems in mind, we focus our discussion on local ABCs and material based ABCs. In particular, we develop in this paper an effective ABC for large-scale dispersive wave propagation problems. Furthermore, we limit our discussion to polygonal domains with straight computational boundaries, and note that the use of circular and curved boundaries (see e.g. [3]) could be an efficient alternative to polygonal boundaries.

The fundamental idea behind most of the differential-equation-based local ABCs is the rational approximation of the exact impedance (or the associated dispersion relationship) of the exterior. Engquist and Majda [4], [5] and Lindman [6] first made use of this idea for unbounded domain modeling and developed a series of ABCs with increasing accuracy, which are henceforth referred to as rational absorbers. Higdon approached the problem from a different direction and devised multi-directional absorbers [7], [8], [9], which can be proven to be equivalent to rational absorbers. Despite the possibility of high accuracy, only less accurate low-order versions of rational absorbers were used, mainly because the higher order ABCs contain higher order derivatives and were difficult to implement in the standard finite element and finite difference settings. In the past decade this situation has improved, with many researchers presenting practical approaches (based on auxiliary variables) to implement high-order local ABCs (see [10] for a review). One important point is that local ABCs require special treatment at corner regions; while several treatments are developed to treat corners [11], [13], they are rather complex involving cumbersome implementation (when compared to material ABCs). Most of the above mentioned researchers focused on non-dispersive waves, except for Higdon [9], who proposed a sequence of ABCs for the dispersive (Klein–Gordon) wave equation, which is recently implemented in a modified approach by Givoli and Neta in finite difference [14], [15] as well as finite element settings [16]. Givoli and Neta also developed similar ABC for dispersive shallow water equations [17].

Material ABCs involve adding a layer with artificial damping right next to the boundary, so that the incident waves decay thus minimizing reflections. The most successful material ABC is the perfectly matched layer (PML) which was originally developed for electromagnetics by Berenger [18], and triggered explosive development of material-based ABCs; we mention few of the developments that are most relevant to the current development. Chew and Liu [19] noted that PML is equivalent to stretching the domain into complex space, a useful analogy that aided further development of PML. Due to the physical and geometrical basis of PML, unlike differential-equation-based ABCs, they are easily extendible to corner regions. The original PML is effective mainly for propagating waves. Extensions to evanescent waves are proposed by Fang and Wu [20] and Berenger [21], [22], [23]. PML is also extended to dispersive wave problems by several researchers [24], [25], [26], [27]. Recently, the PML is compared with rational absorbers by Hagstrom [12], who observed that due to the discretization and truncation errors, PML is less effective than the rational absorbers, but emphasized that PML has better flexibility with respect to treating corner regions.

It was recently discovered that there is a close link between the seemingly disparate PMLs and rational absorbers. Guddati [28] compared the PML with one of the rational absorbers, the continued fraction ABC (CFABC [29]). The link between CFABC and PML are made more precise by Asvadurov et al. [30], who showed that CFABC is related to optimally discretized PML with mixed finite differences. Guddati and Lim [31] later proposed the new derivation of CFABC which links CFABC to special finite element discretization of PML in displacement-based formulation. Based on this idea, they have extended the CFABC to convex polygonal computational domains, making CFABC an effective ABC amenable to standard finite element and finite difference methods. The CFABC, the original version and later derivations, are developed and tested for non-dispersive acoustic wave equation.

In this paper, we extend the CFABCs for dispersive wave equations and test their effectiveness. In particular we extend CFABC to simultaneously absorb propagating waves as well as evanescent waves, which could be predominant when the dispersion is significant. The resulting CFABC is named padded CFABC as it involves a highly efficient way of inserting a padding region that is aimed at absorbing evanescent waves. The padded CFABC, when discretized using finite element method, results in a fourth order evolution equation, which is different from the standard second order equation arising in wave propagation problems. We solve the fourth order equation efficiently by carefully combining Crank–Nicholson method, Newmark time-stepping scheme and operator splitting ideas. The effectiveness of the proposed ABC and its implementation is illustrated using several numerical examples.

The outline of the rest of the paper is as follows. The basic ideas behind the latest derivation of CFABCs are summarized in Section 2. In Section 3, CFABCs are extended to dispersive wave equations (Klein–Gordon equation). Finite element discretization is discussed in Section 4, while Section 5 focuses on the time integration of the resulting evolution equations. Numerical examples are presented in Section 6, and the paper is concluded with some closing remarks in Section 7.

Section snippets

CFABC for straight computational boundaries

Without any loss of generality, we explain the derivation of CFABC for a vertical computational boundary. Essentially, our goal is to replace a full-space with a left half-space and an ABC simulating the effect of the right half-space (see Fig. 1). The procedure entails several steps as described below.

(a)

The first step in the derivation is to discretize the right half-space using an infinite number of finite element layers (note that the discretization is performed only in the direction

CFABC for dispersive wave equation

This paper focuses on the extension of the CFABC to dispersive wave propagation problems. In particular, we consider the Klein–Gordon equation, which is a dispersive equation of the form: - c 2 ( u xx + u zz ) + u tt + f 2 u = 0 , where c is the wave velocity and f is the dispersion parameter. Klein–Gordon equation is considered here due to its simplicity, but it is expected that the proposed ideas can be easily extended to more complex dispersive wave equations arising in several contexts of engineering and

Finite element discretization

Since the CFABC is now based on finite element discretization, no special effort is needed to obtain the finite element mesh (a typical discretization of CFABC with padding is shown in Fig. 5). The padding and absorbing layers on the edge are discretized in a manner consistent with the discretization of the interior domain. The corner region requires no further discretization. It is important to note that (a) midpoint integration is used in the direction perpendicular to the boundary for both

Time stepping

The presence of the W and X terms in (13) makes the evolution equation fourth order in time, as opposed to second order equations in standard dynamics and third order equations resulting from CFABC for non-dispersive wave equations [31]. While it may appear that special effort is necessary to devise time-stepping schemes for (13), the schemes developed for non-dispersive CFABC in [31] could be easily extended to the dispersive case as explained in this section.

Before proceeding to the solution

Numerical examples

In this section, numerical examples are presented to illustrate the performance of the CFABCs with or without padding for the dispersive waves. Due to its computational efficiency we use explicit computation with splitting for all the numerical examples. For all the problems, a Gaussian explosion used in [31] is utilized to generate the dispersive waves, i.e., the forcing function is given by f ( r , t ) = - 2 π 2 f 0 2 ( t - t 0 ) e - π 2 f 0 2 ( t - t 0 ) 2 1 - r 2 a 2 3 , if t ⩽ 2 t 0 , r ⩽ a , 0 , otherwise . In the above, f 0 is the central

Concluding remarks

In this paper we extended and evaluated the continued fraction absorbing boundary conditions (CFABCs) for handling dispersive waves. We observed that, when evanescent waves are not significant such as the low-dispersion case or when the computational boundary is far from the source, regular CFABCs are sufficient. On the other hand, when the computational boundary is close to the source and the dispersion is high, evanescent modes tend to be significant and the regular CFABCs fail. In order to

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. CMS-0100188. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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